Theoretical Considerations and Experimental Probes of the 52 Fractional Quantized Hall State

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Theoretical Considerations and Experimental
Probes of the ?5/2 Fractional Quantized Hall
State

• by Bertrand I. Halperin, Harvard University
• talk given at the
• Rutgers Statistical Mechanics Meeting,
• May 11, 2008
• in honor of
• Edouard Brézin and Giorgio Parisi

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E. Brézin and the quantum Hall effect
From the 2nd paragraph
2.
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?5/2 Quantized Hall State
In 1987, Willett et al. discovered a Fractional
Quantized Hall plateau at Landau-level filling
fraction ?5/2 , the first even-denominator QHE
state observed in a single-layer system. The
nature of this state is still under debate.
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Moore-Read Pfaffian State
Moore and Read (1991) proposed a novel trial wave
function, involving a Pfaffian, as a model for a
quantized Hall state in a half-filled Landau
level. Further clarified by Greiter, Wen and
Wilczek suggested as an explanation for the
quantized Hall plateau at n 5/2 2
1/2. Elementary charged excitations, which have
charge e/4, obey non-abelian
statistics. State is related mathematically to
a superconductor of spinless fermions, with
pxipy pairing. (Described as a pxipy
superconductor of composite fermions.)
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Non-abelian statistics for Moore-Read 5/2 state
Consider a system containing 2N localized
quasiparticles, far from each other and far from
boundaries. Then there exist M2N-1 orthogonal
degenerate ground states, which cannot be
distinguished from each other by any local
measurement. Moving various quasiparticles around
each other and returning them to their original
positions, or interchanging quasiparticles, can
lead to a nontrivial unitary transformation of
the ground states, which depends on the order in
which the winding is performed. ( Unitary matrix
depends on the topology of the braiding of the
world lines of the quasiparticles. Matrices form
a representation of the braid group). If two
quasiparticles come close together, degeneracy is
broken but energy splittings fall off
exponentially with separation.
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Topological quantum computation

Non-abelian quasiparticles may be useful for
topological quantum computation. Kitaev,
quant-ph/9707021 Freedman, Larson, Wang, Commun.
Math Phys (2002) Bonesteel, et al PRL (2005).
Manipulation of qubits would be carried out by
moving quasiparticles around each other, not
bringing them close together. Advantage
exponentially long decoherence times if
quasiparticles are sufficiently far apart.
Caveats 1. Current materials are very far from
this regime. 2. Moore-Read state is actually not
rich enough for general topological quantum
computation. However, this may be possible with
some other states in the second Landau level.
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What is the evidence that the n 5/2 Quantized
Hall State is indeed of the Moore-Read type ?
Evidence comes primarily from numerical
calculations on finite systems.(Morf
collaborators, 2002, 2003 Das Sarma et al. 2004
Rezayi and Haldane, 2000). Using parameters
appropriate to the experimental situation, find a
spin-polarized ground state, which seems to have
an energy gap, and which has good overlap with
Pfaffian wave function. But evidence is not
overwhelming, and is certainly open to
question. Existing experiments do not provide
clear evidence on nature of state. Recent
improvements in quality of materials, and new
experimental techniques,give hope of resolving
these questions.
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What are the theoretical alternatives?
Anti-Pfaffian state Topologically distinct from
the Pfaffian state, but has similar properties,
e/4 quasiparticles, non-abelian statistics.
(Would be equally interesting.) Other kinds of
paired states, including tightly bound pairs in a
fully spin-polarized system, or partially
polarized or unpolarized systems. Would have e/4
quasiparticles but not non-abelian statistics.
(Not so interesting.) Other kinds of quantized
Hall states we havent thought of?
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Anti-Pfaffian State
The Pfaffian (Pf) state is not symmetric under
particle-hole conjugation. The anti-Pfaffian
(APf) is its particle-hole conjugate. Pf and APf
have been shown to be topologically distinct.
(Rezayi and Haldane, 2000 Levin, Halperin, and
Rosenow, 2007 Lee, Ryu, Nayak and Fisher, 2007.
) If you vary the parameters in a system so that
the ground state changes from Pf to APf, there
must be a phase transition separating the two
phases. If the parameters of a system vary in
space, so that one region is Pf and one is APf,
there must be a boundary separating them, with
gapless low-energy excitations. (Both states
have an energy gap in the bulk.)
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Boundary between a Pf or APf ?5/2 state with a
?2 Integer Quantized Hall state (filled Landau
level)
The simplest boundary between a Pf ?5/2 state
and a ?2 state should have two low-energy chiral
modes a bosonic phonon mode and a neutral
Majorana fermion mode, traveling in the same
direction. The edge has a thermal Hall
conductance with K 1 1/2 3/2. The boundary
between an APf ?5/2 state and a ?2 state has a
different structure, and has K-1/2. The thermal
Hall conductance is a topological invariant,
cannot be altered by disorder or boundary
reconstruction. Q ?T K ? ?2kBT/3h
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Edges of a Pf or APf state
?3
?3
K 3/2
K -1/2
APf
Pf
K 3/2
K -1/2
?2
?2
Thermal Conductance K 3/2 1/2 1 -1/2
-1/2 - 1 1
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Nonlinear electrical resistance
Experimentally, it is difficult to measure the
thermal Hall conductance. However, the different
boundary structures of Pf and APf with, say, a
vacuum or a simple ?2 state should lead to
different Luttinger-liquid-type properties, which
should give rise to different forms of
non-linear electrical resistance at a narrow
constriction, which has been studied
experimentally. Recent experimental results
seem to favor APf. (Marcus lab) The
complications expected in a real system have not
been completely sorted out.
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Competition between the Pfaffian and
Anti-Pfaffian State
If one has only two-body interactions, and one
ignores Landau-level mixing, the Hamiltonian of
the half-full Landau level is particle-hole
symmetric. Since the Pf and APf states are
particle -hole conjugates of each other, they
must have identical energies in this model.
Degeneracy can be broken by inter-Landau level
mixing, effects of impurities, sample boundaries,
and deviations from half filling. Sample
boundaries may be particularly important in a
narrow constriction. Pf and APf may coexist, with
a boundary between them. Note The Pf and APf
trial wave functions are exact ground states of
models with three-body interactions, which break
particle-hole symmetry explicitly.
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Proposed experiments to look for non-abelian
statistics, or at least test whether n5/2 state
is Pf, APf or something else.
The most direct demonstration of non-abelian
statistics wold require the ability to move one
quasiparticle around another in a controlled way.
Possible in principle, but we are far from being
able to accomplish this technologically. We
seek other experiments to examine the n5/2
state to see if it is of the Moore-Read type.
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Actual recent experiments
Measurements of the non-linear resistance of a
narrow constriction at n5/2 can give important
information about the state. Interpretation
seems to be complicated Measurements of the
quasiparticle charge. Moore-Read quasiparticles
have charge e/4 . Recent measurements of shot
noise from a quantum point contact at n5/2
support this result. (Heiblum group).
Quasiparticles with charge e/4 are necessary,
but not sufficient could also result from other
states without non-abelian statistics.
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Proposed experiments (1)
. Interference-type experiments directly
related to non-abelian statistics.
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Proposed Interference Experiments at ?5/2
Discussed by Ady Stern and B. I. Halperin (PRL
2006) Other theoretical papers discussing
interference experiments with non-abelian
quasiparticles include Das Sarma, Freedman and
Nayak, (PRL 2005) Bonderson, Kitaev, and
Shtengel, (PRL 2006) Fradkin et al., Nucl Phys B
1998 Bonderson, Shtengel and Slingerland,
cond-mat/0601242 Discuss consequences for
Read-Rezayi parafermion states, possibly
applicable to n12/5 .
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Fix gate voltage at point contacts. Vary area A
by varying voltage on side gate. Measure
resistance V12/I. Expect oscillations in the
resistance as a function of A
1
2


I
I
t1
t2
? 5/2
? 5/2


quasihole
Side Gate
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If n5//2 state is non-abelian Pfaffian or
Anti-Pfaffian state the period of resistance
oscillations should depend on whether the number
of localized quasiholes encircled by the path is
even or odd.
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Weak back-scattering V12??t1 t2 e??i??2 , with
???A B/4?0 , only if the qh number is even.
1
2


I
I
t1
t2
? 5/2
? 5/2


If interference path contains an odd number of
localized quasiholes, quasiparticle path
tunneling at point t2 changes the state of
enclosed zero-energy modes, and cannot interfere
with path tunneling at t1.
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Will these experiments actually work and show
non-abelian statstics?
We dont know. Real systems can be pretty
complicated.
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Acknowledgments
Co-authors Ady Stern, Bernd Rosenow, Michael
Levin, Steve Simon, Chetan Nayak, Ivalo
Dimov. Discussions with experimentalists, too
numerous to name. Financial support NSF,
Microsoft Corporation, US-Israel Binational
Science Foundation.
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Proposed experiments (2).
Measurements of spin polarization. Moore-Read
has complete polarization in second Landau Level.
Measurement should be possible, but difficult at
very low temperatures. (Would be a consistency
check, because some of the alternatives to
Moore-Read are not fully polarized, but not a
definitive test.)
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Existence of a quantized Hall state at ?5/2
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If central region contains an odd number of
localized quasiparticles, this interference term
is absent. Then leading interference term varies
as Re t1 t2 e2?i? 2 . (Period corresponds to
an area containing two flux quanta, rather than
four.)
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Zero-energy modes
Specifically, in a pxipy superconductor, an
isolated vortex, at point Ri , has a zero energy
mode, with Majorana fermion operator gi (from
solution of the Bogoliubov-de Gennes
equations) gi gi , gi2 1 , gi , gj
2dij To form ordinary fermion creation or
annihilation operator need pair of vortices
e.g. c12 (g1 i g2) / 2 ,
c12 (g1- i g2) / 2, obey usual fermion
commutations rules N12 c12c12 has
eigenvalues 0, 1. N12,N34 0 ,
etc. Constraint Number of occupied pairs
Nelectrons (mod 2) . -gt 2N vortices gives 2N-1
independent states
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Explicit relation between Majorana operator and
electron operators
gi ò dr u(r) y(r) v(r) y(r) with
v(r) u(r), localized near vortex. If
vortices are far apart, so there is no overlap
between the wave functions of their zero-energy
states, then these states must have precisely
zero energy. This relates to the fact that
solutions of the BdG equations must occur in
pairs with E1-E2.
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Braiding properties of vortices
Vortices at points R1 R2 R3
R4 .
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Braiding properties
Vortices at points R1 R2 R3
R4 Move vortex 2 around vortex 3 .
Gives unitary transformation g2 g3 . Changes
N12 -gt (1-N12) , N34 -gt (1-N34) .
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Braiding properties
Vortices at points R1 R2 R3
R4 Move vortex 2 around 3 and 4. Gives
unitary transformation g2g4 g2g3 g3 g4
leaves N12 and N34 unchanged. Since vortices are
indistinguishable, get other unitary
transformations by simply interchanging positions
of two vortices. Order of interchanges matter
The unitary transformations do not commute.
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